INVESTIGACI() .....•. I,[VISTA MEXICANA DE F1SICA -111(6) 586....592 DICIEMBRE 2000

z-scan like resuIts produced by linear optical appmximation or a nonlinearmaterial

Rafael Quintero-Torres! and LUL'ila Zamhrann-Valcnci¡¡"2DelJ(/rramento lit' Electrónica. UnÍ\'('rsidm/ AI/ttil/(J/I/{/ Mctmpolit{///{H\z('apotzalc()

Al'. S'an/JaMo # /80, Col. Reynosa '{(¡1I/al/lipas. 02200,\1éxico, D.F, A/('xicoe-IIJail: 1 rquintcro@corrco.a;c./lll/lI.111X: "2!;:'l'@correo.a;;::c.fIlUlI.IIlX

Rosa 1\1aría Ikrmúde/.-CruzDepartamento de GcnéticlI y !Jio!og(a Molecular

CCll1ro de hn'l'Jtif;:acián y Eswdios Al'lllf:lIdos del/n.Hiluto Politécnico NacionalAl'. /lIstiul1n Politécnico Nacional # 2508. 07360 México, D. P. M('rico

('-I1I(fi1:mhe r111((JIgCl/c.cim '('stal'. fIIX

Mrinal ThakurMl'ciJallic(f1 Engil1ecril/g. A/llmm UlliI'crsiry

A//¡'//m Al. 368-19. USAe-mail: mtllllkllr@(.IIX.l1It1m m. edil

Reeihido el 15 de mayo de 20()O: aceptado el 13 de julio dC'2(X)()

A ~l't\llll.tric optics procedmc is lk\'eloped to understand the nonlincar optical cffects of materials. :Imito ohtain .::-scan like results. Thech;lI1~e in lhe rcfractive imk, uue to the optical nonlínearilY is represellteu as a spherieal Icns. Thin and lhick sample cases relativc to (heronforallcnp:th. de/lned by ••n externallcns are modeled. Linear and ll(\nlinear ahsorptíon is illdudcd in matl'rial \I,'ith instan(aneous non linearrCSrt\IISC. slow response mechanisms like thermal effect can he incllldC'd \\'ilh this proccdurc an expbnatioll for the dis(ortion in the z-scanís ohtained wilh lhe increase illlhe aperturc. lhe nonlincar refraclive int!ex ano Ihe sample thit:kness as wdl as \vilh ¡he prescnl'c 01" nonlinear

,lh"'(\l'ptillll

A'cnqmls: :-~can; Ilonlinear refractive index; non linear absorptinn

Se l'mplea ('plica geométrica para entender el efecto óptico no lineal en maleriales y ohlener resultado •..que se asemejan ••1 harrido en .::.Elralllhi{l en d fnJice dc refracción dchiJo a la no linealidad óptica se representa como tilla lente esférica. Se discute el caso de muestrasddp:adas y ~rlles;¡s (el espesor de la muestra se mide respecto a la dislancia confocal definida por la knte eXlerna). La absorción lineal y nolineal :-ocincluyen en materi<llcs con respuesta no lineal instantánea. dc la misma manera"'c pueden considerar mecanismos de respucsta lentaromo los térmicos. Esta interpret~ll'ión da cuenta de la deformación de los resultados dcl harrido en .::al aUlllentar la apcrtur.l. los efe('IOSnolilll'"ks. el e:-opcsory la absorción.

/)C.w"I'il'/orl',\"." Barrido en .:-;índice de refracción no lineal; ahsorciún no lincal

P/\es: 7S20.Ci; -l2.65.-k: 42.70.-a

1. Intrnduction

NonJincar oplics is hecollling one 01' the most imporlanthranches of modern optics dlle lo its trcmendous impact inIhe electronics technology. SOIllC examples 01' appJiLationswhere llo11linear optical phcnolllcllon is playing a mayor roleare frequcney conversion. oplicallllodulation, optical switch~in~. oplical ~ates and optical Illcmories, among others.

Nonlinear optical phcnolllcnon is shown hasically in aHmaterials. Non Iinearity is a property of the ll1edium ratherthan a prorcrty of ligh!. Optiral etTeets an; ovenJri\'cn ¡nlononlinear effects when Jighl 01' iligh inlensity is present inan oplicalmedilllTI. The intcnsity nceded 10 ohserve thesc ('f.fl'els lkpends on the nature (11'Ihe material. \Vhcll ordinaryI¡ghl propagates through an opticall1laterial. lhe elcctric tieldE assoeialeti with lhe liglll heam exerls a polarizing force 011

lhe ch:elrons nI' Ihe material allowing mainly oulcr eleclrons

(o respondo The oplical Illcdiu!ll is polarized and lhis polar-izalion is proportional 10 the applied electric neJd E.

Ir Ihe radiatcd Ileld hecomcs large cnough, lo he com-parahle wilh lhe atomiL lIclds. (hen proportionality hegins tofail and Ihe sllperposilion principIe does not arply any longer.Non linear phenolllcna are dile lo lhe inahilily 01' the dipolcsto resJlond in a linear \Vay \I,:hen light 01' very large ampli-tude interacts with matler. It is eustoll1ary lo express Ihe po-laril.alion P in a powcr series 01' E. wherc ~ rcpresenls thesllsceptihility:

p = ~(II .E + ~i'1 EE + ~(:I):EEE+... (1)

Thc seareh 01" !le\\' and !llore cfticienl materials and dc-vices lo he used in clcclrnnk syslel11s shows continuousgnl\Vlh. The qucst rOl' experimenla! tcchniques to delerminepropcrties, whcrc silllplicity ,lIld lIscfulncss was soughl, pcr-

.:-SCAN L1KE RESULTS PRODUCED BY LINEAR OI'TW:\L API'ROXIMATIO¡-': OF A NONLlNEAR MATERIAL 5M7

WD 1'---.1' ~, ~ focal planeaperturez

photodetector

focus sample

----->

----->

lens

FI(;URE l. z-scan experimental arrangement. the lens defines lh~focus amI lhe sample is scanned through z ncar lhe foclIs.

Fl<itJRE 2. Geometrical t1elinilion.lhe focal plane is defined by Iheexternal Icns. :0 is lhe confocal !cng:th and f is the sample thick-ncss.

,l~<j>xT(:, ~'I') '" 1+ (' aJ(' 1)' (2).r- + .l'- +

The variables used in lhis work are dcfincd in Fig. 2.where lhe sample poSilioll frorn the focal r1anc is ;: = d, -t rD and Ihe oistancc from the fo(."alplane to the aperture isLJ = :; + dl + t. Thc rar lield condition is represented hy13 » =(). (he inclusion of a lhick sample changes de focalplane signilkanlly ano lhe Ilceo 01'an ahsolute th pasitian issolved wilh lhe use 01'lhe h:ns working distance lF D.

apcrturc positianeJ al several limes the confacal length zoaran from Ihe focus. where the con focal lenglh (zo =1rWa/~\) is Jcfined wilh as the minimum spot radius prodllccJhy the focllsing Icns. And measuring the transmission onlywhen lhe samplc is clase lo the focal poi ni. 1::1 ~ '::0. A smallaperturc is lIsed ($ --+ O), .<; indicalcs Ihe percentage of IightIhal passed the aperture. prOllucing lhe nexl sel ol' cquatians.where T slands for tn:msmission as a l'llnclion al' the :: po-sil ion and lhe phasc change .6.(1) due to lhe material, l' is thenormalized POSilioll.~ZI'-I'is the distance hctween the peakamI lIJe valley in lhe ¿-scan signaturc. L\T"_l' is the transmis-sion dilTerence het\veen lhe peak ano the valley in the ;:-scan:

(4)

(5)

(3),1'

mittetl the tlevelopment 01'lhe so called z-scan Icchniqllc [11.llscd lo Illcasurc non linear refraclivc index ami nonlincar ah-sorplion. In Ihis lechnique. a lhin sample is scanned throllghlhe fOCllS01'a gallssian laser beam. where ¡¡ghl can pass fromsitie lo sitie 01' lhe sample with small Jislortion 01' lhe wavefront. A sensor is placed hehind a lInile aperturc in lhe far¡¡eld 10 monitor lhe output 01'the sample (Fig. 1). The exper-imental arrangement is se! in a way lhal a single hcam anda single palh 01' light is llscd lo delermine the nonlinear re-fraclive index and Ihe non linear ahsorplion. II is possihle loachievc lemporal resalution if a Mil:hclson inlerferomeler isuseJ hcfore lhe .::-scan arrangell1ent lo produce one path-lwoheams 12J experil1lenl. using lhe delay arl1l in lhe inlerferom-eter lo produce autocorrelatian. constanl average power amivariahle peak power.

Slarting lhe scan near the lens (far fmm focus). the inlen-sily is low anJ lhe transmittancc lhrough lhe aperture is con-stan!. As lhe sample comes nearer lo lhe focus. the intensityhecoltlcs largcr and a self-foclIsing effecl is ohserved (case01'posilive Ilonlinear refraclive index). It rnakes Ihe hearn locollirnate resulting in an increase in lhe lransrnittance. \Vhenlhe sarnple goes heyond lhe focal plane, a self-defocllsing oc-curs leading: a heam broadcning at Ihe apcrturc and Ihe lrans-millance tleCfcases (lhe apposile is ohlaincd when materialused has a negalive nanlinear refractive index). Similar con-clusions have heen rcacheJ for lhe z-scan signaturc. whendifferenl consideralions were laken [3,41.

2. 'l,-sean rcsults

In lhis section. we prescnt Ihe rcsults ohtained with the ::;sean leehnique and it is campared wilh resulls oblained \\'hen(he geomelric optics techniquc is llscd. In lhe original .::-sean work, a simplification 01' Ihe nonnalil.cd on-axis .::-sean lransmittancc is derived considering small nonlincari-tics, lil(I'1 « 1, where the rhase change for Ihe wave aflerlhe material (~(l) = '!:rru'lJof / >') is tlefined hy lhe nonlinearrcrractioll index 112, Ihe inlensily lo. lhe lhickness t and lhe\•..avclcnglh lIsctl .\. A far f1eld condition was llsed. wilh Ihe

3. GcomctrÍC"optics approximation

T\\'o general cases are considercd in Ihis wark; in the lIrstcase, a lhin sample 01' non linear oplical malerial is modeledas a linear optical elcmcnl wilh properlies dependent on lheinlcnsily and in lhe second case. Ihick sJmples are motleledas a set of sIices. cach OIlC with a fraclion of lhe elTeet. Inholh cases. lransparcnt nI' ahsorhing malerials are aSSl1l11co.In this scclion we descrihe lhe sleps 01'lhe procedurc lhal areCOIllJllOlllo hoth cascs.

R,,', Me.\', Fú, 4(, (6) (2(XXI),~6-,92

5XX R. QUINTERO-TORRES, L. ZAMBRANO-VALENClA. R.M. BER~lUIJEZ.CRUZ. ANIJ M. THAKUR

( 12)

(1 1)

( 10)

(13)

n1M

R = a + 1Mn

,.

( D )c+ --(ji".''''B .

\+(-)(j'''PlIt

I

lA /JI'ollt.pnt = e n lIIput,

IIWJ [1 + (zzJ ']

(~) ~'I'[1+ e)']'.11 -o ....0

1_(n-1)¡=-R-

where

In order to determine the change in Ihe complex heam ra-dius q(::) al any point as a function of any oplical elemcnl inIhe path, Ihe use of an Al1C D matrix or transfer matrix \\'illhe llsed:

F[(iURE J. Contrasling a gaussian amI a spherical funclion, lo re-lat(' w and R.

(H)

(7)

(6)1R(z)

1q(z)

2P [(Z)']-1lax', = _ 'e') = /0 1+:- .,,:..; •.. ...0

[ 2"']1(,.. z) = la,', exl' - w'(z) ,

whcrc R(::) is the \Vavcfronl radius 01' curvaturc and w(.::) isthe.:hcam radius.

The hasic qucstion lO 501\'c is: haw does Ihe gaussianhcam q(;;) changc as it passcs through a non linear plale?In otller wonJs. hO\\' Jocs lhe lransmission through an apcf-lun: al far lIcld changc whcn Ihe samplc movcs ncar lhe fm:aiplanc'!

The gcolllclric optics approximalion tcchniquc is hascd011lhe assumplion 1ha1 a changc in (he rcfractive indcx takcsplace (11 + ~1I) \\'hcn a Jaser hcam (sourcc al' high-intcnsityl"(lherent light) impinges 011 a nonlinear material. This changetille 10 lhe light intensity is Jirectly related to the eleclronicinteraction or indirectly related to thermal effects and fol1owsIhe intensity dislrihution 0'- the ligllt, in this case a gaussiansllape or lhe temperature distrihution. Considering the trajec-tmy 01"the light, it is equivalent lo considcr a parallel platewilh ;¡ rel"raclive indcx function of lhe position z ami 1', or¡¡ malerial wilh a unifofm rcfraClive index n with a non par-<llIel plate, rathef a gcomclrical shape resembling lhe plateplus ;¡ gaussian deformation. Idcally, a gaussian lens shouldbe added or subtracled to the samplc. howcver the simplicilyul" Ihis approxilllation consisls in the use of a spherical len sinstead,

Thc inlensity distribution ror Ihe Gaussian bealll al radiusl' antl posilion ':, 1(1',.::) is reprcsented in the next cquatioll,\\'Ilcrc lhe intensity at focus, r = O alllI z = ()is lo:

In the dcrivation of the gcomctrical optics techniquc, thecumplex heam radius [5.51 q(z) is used lo sludy lhe propaga-(ion 01' the gaussian beam through a nonlincar medium. Thisradius is rcprcsented by (he known cxpress ion

The focal length oí" a Icns is delineo as the radius of Ihelens divided hy the change in the rcfractivc index prodllccdby lile Icns:

Suhscquenlly. finding an equalion for the focal lenglh 01'the spherical Icns as a fUl1clion 01"lhe nOl1lincar properticsin 111t:material and the inlensity dislrihlltion w(z), is rossi~ble, Following Fig. 3, lets assulllc Ihal al .,. = w the sphereIOllcht:s (he sample ami also the radius of the sphere is biggclIhan Ihc Ihickness 01' the salllple R > tón/n, meaning IhatIhe nonlincar cllcct is small, From Ihal it is easy to reach ancxpression l"orR:

where lhe matrix elelllcnts A, EJ,e, and D havc infonnatinn01"all optical clclllcnts hctwccn the inpul and lhe outPlll.

Next equation is a general cxprcssion for the change inlhe lascr spot size with Ihe inronnalion of Fig. 2:

The linear transmission through an aperture ol" radius (l.

T,.; is dellned as the ratio 01"Ihe powcr carried wilhin lhe cir-cle of radius (1 in the trans\'erse planc at position z, P(a),to Ihe lotal power P(r -+ N). In the linear case, it is theconstant ,<;:

( 15)

( 14)

[ 2"']I - ,"xI' - wf.(z) ==-'.

., (I! -.-\ WD)'[..\, + ]

,) ,) ':0w~,,,= ,,"'o .-\D _ Be .

'1', (z) = P(II). P(/' -+ 00)

(9)w'(z)1l

R '" --~--.2/Il,/(,., z)

RCI', MI'X, Fú .. 1(. (6) (20(X» 586-592

,.SCAN L1KE RESULTS PRODUCED BY LINEAR OPTICAL API'ROXIMATION 01' A NONLlNEAR MATERIAL 589

FIGURE 4. Typical z-scan transmission through difTercnt apcrturesH. Itere the origin of rhe asyrnmetry in the z-scan is explicable.

( 19)dI

1- -J

d,1- f

1

J

input I t- dI -+ O t- d, -+ loutput,

I~~I=

The expression of the A Be D matrix for a nonlinear sys-tem is

in this case

indicates the need to use two propaga(ions dI, d2 and (he non-linear erreC( represenled hy only one ideal variahle Ihin lens(O).

To obtain the transmission cquation, we suhstitule the lasl(wo AlJCD cqualions in Eq. (14) in order to oblain the laserspot sil.c for a linear ami a nonlinear system respectively, amiIhen subslituting in Eq. (17) lo end up with:

2

_, ._.J

1.51T.0.5

-s=O---$=0.1. 5=0.5_ s=0.9

O

2

.8

.6••.2

•.• 1

0.8

0.6

OA0.2

O

Finally. T defines Ihe transmission as the ralio hClwccnlhe powcr through lhe apCrlurc duc lo a nonlinear systclll dc-f1ncd hy an ABCD rnatrix ano a linear systcm dcflncd hyolhcr A13CD matrix. (T..•a/TL. this considcration willmakelhe Iransmission cqualto one whcn a linear material is uscd").

4. Thin lransparenl sample analysis, Ihicknesssmaller Ihan Ihe confoeallenl:lh, t < Zo

Figure 4 shows Ihe aperlure elleel IEq. (16)J, anu inui-cales Ihal \\'hcl1 ~ --+ 1 (Iarge apcrturc) Ihe z-scan transmis-sinn grcalcr Ihan one goes raster (o 1 than Ihe z-scan trans-mission smaller than one: that is. the aperture will affect lhesymmctry in .:-scan, shrinking Ihe positive hump faster thanlhc negativc onc. This information is uscful because manysolids are no! perfectly homogcneolls and a larger apcrture isIlcclh:d lo produce a Illeaningflll reslIll.

(20)

(21 )

(22)

(23)

Z;f::::: -.

':-0

Z5+(z+d,)'

[=0 (1 - Ji) r + [d, + z (1 - 7)r("-1) 2~<I>x

1"(z, ~<!» '" 1 + -,,- (.r' + 1)"

Simplirying this cquation with the rar field condition,d, » :0 anu Ihe express ion for Ihe foeallength [Eq. (10)1,will produce the set 01' expecled cquations:

(16)

( 17)

T = TVL(,,)

- Tc(,,)

'1" - l' T wZ(z)= 1111 = ~( J'

.1"-"'40 w....•.L Z

illput [+- di -tt- d'2 -t [Ollt)lut.

1\ scl of cquations for typical .:-scan condilions where a. thinsamplc with t « ':0 and intensity small enough such thal~(I) « 1, can he obtaincd if the AlJCD matrix for the Iin-car systcm and for Ihe nonlincar systclll are oefined and fromthclll Ihe W""l is dclermined.

Thc ..lIJC D matrix that characteril.cs a linear systempropagating through air with

is c.'{pressc~

dI + d'l1 .

when: only (wo propagalions di and d'2 are needed.

( 1X)

5. Thick samples analysis, Ihickness larger IhanIhe confocal lenglh (1.> zo), wilh linear andnOlllinear ahsorplion

This case is a generalization 01' Ihe thin samplc case whercf > Zo and or .6.4) » O. In this proccdure, mulliplc nonlin-car Icnscs replace the single nonlinear lens to kcep Ihe localsilualion inside Ihe scope 01' lhe previous case. The amountof lenses is defined oy Ihc thickncss and the intensity. Nowa nc\v situalion is taken i!llo accounl, the powcr ¡nside Ihes<lmplc dccrcases duc to linear and nonlinear absorp(ion. (hisabsorption is iI1lensily-dcpendent.

For Ihis case the inlcnsity is nOI a good numbcr becauscof the variation in Ihe spot arca, Ihcrefore a betler one is thepowcr (P) and the general cquation lo find Ihe power is a

Rel'. Me.\". Fú. 46 (6) (2000) 5R6--592

5911 R. QUINTERO-TORRES. L. ZAMBRANO-VALENCIA. RM BERMÚOEZ-CRUZ. ANO M. THAKUR

fUlletion of lhe linear absorption (l and Ihe non linear absorp-lion 13. whcrc f3I(u) is Ihe changc in absorption tlue lo lheinIensity (~'l):

in lhe rtIdius wilh an exponcntial variation as follows:

{(

b ( )2 '" '" exp( -klc + "J). (27)+ e+ u

dP-1 = -[o + ¡J1(1l)jP., u

(24) When Ihe arca is uniform (k = O), Ihe resuh for absorp-tion in a wavcguide is recovcrcd 17]:

(28)\\fhefe lhe powcr ¡nside lhe samplc as a function of inter-

nal Icnglh 1 is

11mP( )

= T Po exp(ou) - -- exp(oll)11 10 a+p

x (exp( -Oll - k[c + uJ) - exp( -kleJ)J, (25)

1(1) = exp('w) {~[1- ,'xp(-ou)J + _,_1_}.

lt o: 1./0

For lhe thick film, Ihe nonlincar material is suhstituted hya set 01' slices 01' the same material, where each ooe can hemodeled as a "non linear leos". The explicit ABCD matrixelcmcots defining a linear thick system

2137/1= --2'

JrWn

iUJlut I <-- d, ---+ 1II <-- d, ---+ 101ltpllt,

whcrc 'JI, TI. nl represen! rcfractivc indexo incident surfacclransmiss;on amI incident power rcspcctivcly. Tú end up withan elemental fUllction, it was nceded lo change lhe varialion

I

where 1'1 rcpresenlS the samplc thickness.The Ilonlinear system over lhe samc path is schematically

represcnlcd wilh

iup"1 I <-- di ---+ [JO[ JO ... [)O <-- d, -> IOlllplll.

e = (rI - Il' D)Il,

/,-= ~ In (1 + ~) ,2 lI~':O

(26)

is

1

.4 EI_11C D - O

dl+d2+11 n

1(29)

and1

1

Ji

Jn

J1--

f.H

J1(

J1---

Ji-IU

1

1f¡

J1(

J1--

f¡1!

I~'ill (30)

all optical clclllents are includcd. 11 reprcsents a slicc 01' linear material and O rcprcsents an ideal nonlinear lens associated

with that sIice.The ABC D elcltlenls for the Ilonlincar part are ohwined irom mulliplicalion of 311 matrices. The cxplicit forms for each

A, n. e ami D ror 11I elements are shown in Eq. (31) where i gocs from 1 to '111.

[J = A.dl + [J"

[Ji = Am-'+I (f) + Bm_'+I,

(31 )

Dm-i+l+ J 'm-i+1

C = C"

D = C.dl + DI'

Di = Cm-'+I (f) + Dm-i+l,

Do = 1.

Ca = O,

( J) Bm-'+I1- --- +---,nJm-i+l Jm-i+1..ti:=: Am-i+1

An = 1,

To implcmcnt those expansions is simple, and a seqllenccis olltlined next:

• From the original conditions define the amollllt 01'elcll1cnls to use. If t > Wo the numhcr of cle-mcnts is t/wo and the slicc thickncss 8 :=: (samplclhickness)/(number of elements).

• Calclllate P at each clemcnt using Eq. (25) as a func-lion 01' di. di is now a hctter refcrence than z :=: Ohccausc il is absolutc. The change in z :=: O rclative tolhe physical equipment is due to a focal planc changeas the sample moves around il. However Ihis shift canhe calculated at any particular situation.

Re" Mex. fiJ. 46 (6) (2000) SR()-Sn

,-SCAN LlKE RESULTS PRODUCEn BY LINEAR OI'rJCAL APPROXIMATION OF A NONLlNEAR MATERIAL 591

0.83

0826

0828

0829

0825

\00

--------_ 0831

0824

----------------10823150

, 004

1003

1002

100' I':~ -p0999

T0998

...... TllUl

OW71 T.0996

-'50 -'00 -50 o 50z(j.lmt

TA"'-E 1. Paramelers used in the simulalion of Figs. 5 amI 6.

>. 1.06411m o 0.6 cm I

d'l lOO =0 (3 0.05 cm/MW

11 1.86 1'0 5W11] -1 x lO-s cm2/MW s 0.5

=0 47.24¡lm Wo 4¡lffi

o Calcula le f for each ciernen! Eq_ (10)_

o Calcula!e ABe D using Eq_ (31)_

o Calcula!e w's using Eq_ (14)_

• Calculate T..\ [Eq. (17)]. it is identified with !he Asuhindcx bccausc now the absorption crfcet is incJudcuin Ihe focal Icngth calculation. Calculatc lhe (otalpowcr lransmittcd by lhe samplc (T p). Ihal mensuresIhe transmission without lhe apcrturc (B -+ 1). and ilis lhe ratio hClwccn total output powcr (Pou•v''') and in-put powcr (Po). The normalizcd lransmission throughIhe samplc in a z-scan cxpcrirncnt is Ihe lIlultiplica-{ion ofT", ano Tp; and this is called Ihe real transmis-sino (T"EA¡J. (TnEAd rcprcscnls the outpul sigll<ll in<lnactual z-scan experiment with <inaperture (uscLl toI11easurenon linear rcfractive inLlcx);Tp rcprcsents theOUlput signal in an actual z-scan experimellt with noaperture (useLl to measurc nonlinear ahsorption).

Tahle J ineluLles the paramelers(R) useLlto ohlain Figs. 5ami 6. Pigure S show T. (5 -> O), the ideal z-scan transmis-sion rellloving the aperture elTect and the linear antl nOlllin-ear elTect; T(s ::::::;0.5), the z-sean translllission 1lI0dilietl hyIhe apenure effeel; T"BAd" = 0.5, (1 i O, fI i O) repre-sents an expeeted actual experimental z-scan results lo scethe refractive index and Tp(s -+ 00, n i O, iJ i O) rep-rescnls an expcctctl aClUal experimental z-scan results lo seethe nonlinear ahsorption. AlI these variahles were calculatetlin samples with a Ihicknessof 10 11m, 100 IIIll anLl600 IIIll re-speclively. From them it is possihle to see lhe thickness efreel(hecause Zo ......,50 11m), the efrect due to ahsorption anLllheconsequence of the aperture sizc. T. ano Tare cqual lo ollewhcn the sample is far from the high inlcllsity lone. The lin-ear ahsorption in the Illaterial L1ccrcasesTUEAL valucs and theoverall efreet, as was expcctcd. The Illagnitutle 01" the Ilonlin-earilies measurcd hy pl~ak lo valley lransmission is smallcr inthe lhin sample. Figure Se shows that a Ihick samplc (1 > zo)will have lhe distanee betwcen peak antl valley position re-laled lOlhe thiekncss (.6.ZI'_t' ......,I/n) ratha Ihan lo the COI1-

foealleng!h.

(a)

1.03 0855

102 0845

1,01 0835

~~ 0,825 ¡.: -T' ~ow T 0815

.. T.~098 0805T.

0,97 0795.200 -150 -'00 -50 o 50 '00 150

z(f.lm)

(b)

0875

086!

O,85~

0845

0835 .:

~ 0825 ¡~OW 0815 ••

098 0805

097 0,795

096 0785

095 0.775-460 -360 -260 -160 -60 40 140

:r:(f.lm)

(e)

FIGURE 5. a) z-scan Iransmission for a sample wilh a thickness of(a) 10 ¡lln, (b) 100 ¡1m and (e) 600lim, and a eonfocal Icngth of.t7.2..t 11111.

(32)- T PO"II'''I - T l'THI~AI. - A Po - A 1"

• Finish the sean. changing di.

Rrl' Mex. PÚ. 4~ (6) (2()()()) 586-592

592 H. QUINTEHO-TORRES. L. ZAMIlRANO-VALENCIA. R.M IlEHMÚDEZ-CHUZ. AND M. THAKUR

FIGURE 6. Thrcc plots fm the chang:c in Iransmission as a funclionof the thickncss. ideal (T.). apcrturc !i :;:::0.5 (T). anel absorptiondiffefcnt from cero (TH.EAd. One plot for Ihe changc in Ihe dis.lance bctwccn peal and vallcy. the rcsult is approximatcly the samefor the Ihrcc cases.

Figure 6 shows fouT plots as a function 01' samplcthickncss, fmm 10 times srnallcr than (he con focal Icngthlo 100 times the confocal Icngth. ~Tl'-t' ITlcasurcs themaximurn lransmission T,><..•k. minus minimum transmissionT

v"Il •.).( .•.•.-1 O). Similar ucflnitions stand ror T.t....= 0.5),

ano for TUEAI.(S = 0.5.0' -:f; D,/j 1= O). Hcrc il is possihlcto sec linear ¡nerease of lhe phasc changc as (he thicknessincreases and saturales for thickncss largcr than lhe confo-

0.1

!;j 001

0.00101

100

"'.'0

,10 ~..,

~ T'

~T

" ~ T,

~,1

10 100

cal length. For TREAL lhe confocallength decreases when thelhickness inereases due to the ahsorplion or if lhe incidentpower is high enough. In lhe righl Y -axis lhe changc in z dis-tanee hetwcen pcak and valle)' is shown as a function of sam-plc lhickness. For thickness larger than the confocallength alinear increase can he seen, for thin samples this is a constantvalue and proportionalto the confocallength. The differencehelween slandard (t « =0) and eXlended (t > zo) z-sean isclear.

6. Summary

A simple rcvicw 01' classical oplics is useful lo describe apO\\'crfullechniquc wilh widesprcad use lo characlcrizc non-linear malcrials. Thc well known results are reproduced andgencralizalions toward Icss common conditions were pro-vided; Ihey are userul mainly in solids where lhe samplcconditions or lhe laser characteristics are not always undercontrol. An added gain is the insight in the experimental ar-rangemenl than can be obtained by analyzing the geometryeffccts. This work is iIIustraled with simulated resuIts froma nonlinear material to discuss the interrelation between thetechnique arrangements. lhe experimental resuIts and the ma-terial properties.

Acknowlcdl:ments

This projcct was supported by a grant from National Council01' Seience and Teehnnlogy fmm Mexieo (CONACyT), underprojeeI29032-A.

1. ~1. Sheik.Bahac et al .. ¡EEE J QfI{lIItum Elt'ctnmics 26 (\1)90)760. and rcfcrclKcs thcrcin).

:2. R. Quintero-Torres ami M. Thakur. AI'I'1. Phys. L('ll. 66 (1995)1310.

3. M. Shcik-Bahac l'f al .. 0l'tical EIIg;,u'l'rillg 30 (1991) 122X.4. lA. IIcrmann and R.G. McDuff. 1. Opt. Soco Am. /l. tu (1993)

2056.5. A.E. Siegman. UI.H'rs. (University Scicncc Books. Mili Vallcy.

Calif.,1986).

(,. A. Yariv, 0l'linl/ Dl'ctrollics, (Hoh. Rinehan and WinslOn.Ncw York. 1(76).

7. L.A. f1ornak, Polymers for Lightwa\'e and ¡IIregrated Optic.cTl'C/lfIologv (lTU/ Al'plicatirms. (l\lareel Dekker, Ine, New York,1992).

8. R. Quintero-Torres and M. Thakur. 1. Appl. Phys. 85 (1999)401.

R('I'. Mex. F{t . ..¡r. (6) (l(X)O) 5X6-592